SOLUTION: Two circles have perimeters that add up to 12π centimeters and areas that add up to 20π square centimeters. Find the radius of each circle. Give exact answers.

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Question 229541: Two circles have perimeters that add up to 12π centimeters and areas that add up to 20π square centimeters. Find the radius of each circle. Give exact answers.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the radius of the first circle
Let y = the radius of the second circle circle

Since perimeter (better known as circumference) of a circle is 2pi%2Ar and area is pi%2Ar%5E2 we get the following equations:
2pi%2Ax+%2B+2pi%2Ay+=+12pi
pi%2Ax%5E2+%2B+pi%2Ay%5E2+=+20pi

Since the second equation is not linear, we'll use the Substitution Method to solve this system. We'll solve the first equation for y. Start by subtracting 2pi%2Ax from each side:
2pi%2Ay+=+12pi-2pi%2Ax
Divide both sides by 2pi:
y+=+6+-+x
Now we'll substitute this into the other equation:
pi%2Ax%5E2+%2B+pi%2A%286-x%29%5E2+=+20pi
Simplifying. We'll get rid of the pi's by dividing both sides by it:
x%5E2+%2B+%286-x%29%5E2+=+20
Since this is a quadratic equation we'll simplify and get one side equal to zero:
x%5E2+%2B+36+-+12x+%2B+x%5E2+=+20
2x%5E2+-12x+%2B36+=+20
2x%5E2+-+12x+%2B16+=+0
Now we solve this by factoring or by using the quadratic equation. This will factor fairly easily:
2%28x%5E2-6x%2B8%29+=+0
2%28x-4%29%28x-2%29+=+0
By the Zero Product Property we know that this product can be zero only if one of the factors is zero. The factor of 2 cannot be zero but the other two factors can:
x-4+=+0 or x-2+=+0
which gives us:
x+=+4 or x+=+2

Remember to answer the question. (With word problems it is always tempting to feel satisfaction with having solved for the variable and forget to answer the question.

The question here is to find the two radii. Since x is the radius of one circle, we need to find y, the radius of the other circle, too. We can get y using our x values and the equation y = 6 - x.
For x = 4:
y = 6 - 4 = 2
For x = 2
y = 6 - 2 = 4

So even though we came up with 2 x values, there is only one solution. The radii are 2 and 4.